Question: Amir stands on a balcony and throws a ball to his dog, who is at ground level. The ball's height (in meters above the ground), $x$ seconds after Amir threw it, is modeled by $h(x)=-(x+1)(x-7)$ What is the maximum height that the ball will reach?
Solution: The ball's height is modeled by a quadratic function, whose graph is a parabola. The maximum height is reached at the vertex. So in order to find the maximum height, we need to find the vertex's $y$ -coordinate. We will start by finding the vertex's $x$ -coordinate, and then plug that into $h(x)$. The vertex's $x$ -coordinate is the average of the two zeros, so let's find those first. $\begin{aligned} h(x)&=0 \\\\ -(x+1)(x-7)&=0 \\\\ \swarrow &\searrow \\\\ x+1=0\text{ or }&x-7=0 \\\\ x={-1}\text{ or }&x={7} \end{aligned}$ Now let's take the zeros' average: $\dfrac{({-1})+({7})}{2}=\dfrac62={3}$ The vertex's $x$ -coordinate is $ 3$. Now let's find $h({3})$ : $\begin{aligned} h( 3)&=-( 3+1)( 3-7) \\\\ &=-(4)(-4) \\\\ &=16 \end{aligned}$ In conclusion, the maximum height that the ball will reach is $16$ meters.